The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order

TL;DR

This paper proves that Weyl alternation sets are order ideals in the weak Bruhat order for classical Lie algebras, characterizes these sets for sl_{r+1}(\u00C4), and explores their enumerative properties related to Kostant's formula.

Contribution

It establishes that Weyl alternation sets form order ideals in the weak Bruhat order and provides a complete characterization for sl_{r+1}(\u00C4), answering a recent open question.

Findings

Weyl alternation sets are order ideals in the weak Bruhat order.

Complete characterization of Weyl alternation sets for sl_{r+1}(\u00C4) with highest root EDF1F3n.

Enumerative results supporting future conjectures on the q-analog of Kostant's weight multiplicity formula.

Abstract

For integral weights $\lambda$ and $\mu$ of a classical simple Lie algebra $\mathfrak{g}$, Kostant's weight multiplicity formula gives the multiplicity of the weight $\mu$ in the irreducible representation with highest weight $\lambda$, which we denote by $m(\lambda,\mu)$. Kostant's weight multiplicity formula is an alternating sum over the Weyl group of the Lie algebra whose terms are determined via a vector partition function. The Weyl alternation set $\mathcal{A}(\lambda,\mu)$ is the set of elements of the Weyl group that contribute nontrivially to the multiplicity $m(\lambda,\mu)$. In this article, we prove that Weyl alternation sets are order ideals in the weak Bruhat order of the corresponding Weyl group. Specializing to the Lie algebra $\mathfrak{sl}_{r+1}(\mathbb{C})$, we give a complete characterization of the Weyl alternation sets $\mathcal{A}(\tilde{\alpha},\mu)$, where $\tilde{\alpha}$ is the highest root and $\mu$ is a negative root, answering a question of Harry posed in 2024. We also provide some enumerative results that pave the way for our future work, where we aim to prove Harry's conjecture that the $q$-analog of Kostant's weight multiplicity formula is $m_q(\tilde{\alpha},\mu)=q^{r+j-i+1}+q^{r+j-i}-q^{j-i+1}$ when $\mu=-(\alpha_i+\alpha_{i+1}+\cdots+\alpha_{j})$ is a negative root of $\mathfrak{sl}_{r+1}(\mathbb{C})$.